After installation, I used the hardware driver manager to install flgrx-updates. After rebooting, I get to grub, select Mint, and then it shows the splash screen with the green LM logo on it. It stays on this screen forever and never starts the desktop.

It's not frozen -- I am able to hit ctrl-alt-delete to reboot it. However, it never progresses from this splash screen.

Use xenopeek's advice here, Solve boot problems with AMD and NVIDIA graphics cards, during the next reboot in order to disable the "quiet splash" mode.
As a result you should read a lot of messages during the startup process.
At the point in time where your splash screen would normal simply stay on the screen forever, you should now be able to read (error) messages which explain why the splash screen never goes away. Very likely something about being unable to load the graphical desktop environment.
We are curious to read these messages, too.

Kind regards,
Karl

Old bugs good, new bugs bad! Updates are evil: might fix old bugs and introduce no new ones.